# Universal representation (C*-algebra)

In the theory of C*-algebras, the universal representation of a C*-algebra is a faithful representation which is the direct sum of the GNS representations corresponding to the states of the C*-algebra. The various properties of the universal representation are used to obtain information about the ideals and quotients of the C*-algebra. The close relationship between an arbitrary representation of a C*-algebra and its universal representation can be exploited to obtain several criteria for determining whether a linear functional on the algebra is ultraweakly continuous. The method of using the properties of the universal representation as a tool to prove results about the C*-algebra and its representations is commonly referred to as universal representation techniques in the literature.

## Formal definition and properties

    Definition. Let A be a C*-algebra with state space S. The representation
       $\Phi :=\sum _{\rho \in S}\oplus \;\pi _{\rho }$
    on the Hilbert space $H_{\Phi }$ is known as the universal representation of A.
As the universal representation is faithful, A is *-isomorphic to the C*-subalgebra Φ(A) of B(H$_Φ$).

### States of Φ(A)

With τ a state of A, let π$_τ$ denote the corresponding GNS representation on the Hilbert space H$_τ$. Using the notation defined here, τ is ω$_x$ ∘ π$_τ$ for a suitable unit vector x(=x$_τ$) in H$_τ$. Thus τ is ω$_y$ ∘ Φ, where y is the unit vector Σ$_{ρ∈S}$ ⊕y$_ρ$ in H$_Φ$, defined by y$_τ$=x, y$_ρ$=0(ρ≠τ). Since the mapping τ → τ ∘ Φ$^{−1}$ takes the state space of A onto the state space of Φ(A), it follows that each state of Φ(A) is a vector state.

### Bounded functionals of Φ(A)

Let Φ(A)$^−$ denote the weak-operator closure of Φ(A) in B(H$_Φ$). Each bounded linear functional ρ on Φ(A) is weak-operator continuous and extends uniquely preserving norm, to a weak-operator continuous linear functional ρ on the von Neumann algebra Φ(A)$^−$. If ρ is hermitian, or positive, the same is true of ρ. The mapping ρ → ρ is an isometric isomorphism from the dual space Φ(A)$^*$ onto the predual of Φ(A)$^−$. As the set of linear functionals determining the weak topologies coincide, the weak-operator topology on Φ(A)$^−$ coincides with the ultraweak topology. Thus the weak-operator and ultraweak topologies on Φ(A) both coincide with the weak topology of Φ(A) obtained from its norm-dual as a Banach space.

### Ideals of Φ(A)

If K is a convex subset of Φ(A), the ultraweak closure of K (denoted by K$^−$)coincides with the strong-operator, weak-operator closures of K in B(H$_Φ$). The norm closure of K is Φ(A) ∩ K$^−$. One can give a description of norm-closed left ideals in Φ(A) from the structure theory of ideals for von Neumann algebras, which is relatively much more simple. If K is a norm-closed left ideal in Φ(A), there is a projection E in Φ(A)$^−$ such that
    $K=\Phi (A)\cap \Phi (A)^{-}E,K^{-}=\Phi (A)^{-}E$
If K is a norm-closed two-sided ideal in Φ(A), E lies in the center of Φ(A)$^−$.

### Representations of A

If π is a representation of A, there is a projection P in the center of Φ(A)$^−$ and a *-isomorphism α from the von Neumann algebra Φ(A)$^−$P onto π(A)$^−$ such that π(a) = α(Φ(a)P) for each a in A. This can be conveniently captured in the commutative diagram below :
Here ψ is the map that sends a to aP, α$_0$ denotes the restriction of α to Φ(A)P, ι denotes the inclusion map.
As α is ultraweakly bicontinuous, the same is true of α$_0$. Moreover, ψ is ultraweakly continuous, and is a *-isomorphism if π is a faithful representation.

### Ultraweakly continuous, and singular components

Let A be a C*-algebra acting on a Hilbert space H. For ρ in A$^*$ and S in Φ(A)$^−$, let Sρ in A$^*$ be defined by Sρ(a) = ρ∘Φ$^{−1}$(Φ(a)S) for all a in A. If P is the projection in the above commutative diagram when π:A → B(H) is the inclusion mapping, then ρ in A$^*$ is ultraweakly continuous if and only if ρ = Pρ. A functional ρ in A$^*$ is said to be singular if Pρ = 0. Each ρ in A$^*$ can be uniquely expressed in the form ρ=ρ$_u$+ρ$_s$, with ρ$_u$ ultraweakly continuous and ρ$_s$ singular. Moreover, ||ρ||=||ρ$_u$||+||ρ$_s$|| and if ρ is positive, or hermitian, the same is true of ρ$_u$, ρ$_s$.

## Applications
### Christensen–Haagerup principle

Let f and g be continuous, real-valued functions on C$^{4m}$ and C$^{4n}$, respectively, σ$_1$, σ$_2$, ..., σ$_m$ be ultraweakly continuous, linear functionals on a von Neumann algebra R acting on the Hilbert space H, and ρ$_1$, ρ$_2$, ..., ρ$_n$ be bounded linear functionals on R such that, for each a in R,
    $f(\sigma _{1}(a),\sigma _{1}(a^{*}),\sigma _{1}(aa^{*}),\sigma _{1}(a^{*}a),\cdots ,\sigma _{m}(a),\sigma _{m}(a^{*}),\sigma _{m}(aa^{*}),\sigma _{m}(a^{*}a))$
    $\leq g(\rho _{1}(a),\rho _{1}(a^{*}),\rho _{1}(aa^{*}),\rho _{1}(a^{*}a),\cdots ,\rho _{n}(a),\rho _{n}(a^{*}),\rho _{n}(aa^{*}),\rho _{n}(a^{*}a)).$
Then the above inequality holds if each ρ$_j$ is replaced by its ultraweakly continuous component (ρ$_j$)$_u$.